Question

Find the number of integers between 1 and 500 that are not divisible by any of the integers
2,3 and 5

Answer 1

Answer:

500/2 = 250 No. of integer divisible by 2 is 250. ∴ n(A) = 250 5000/3 = 16666 ∴ No. of integer divisible by 3 is 166. i.e., n(B) = 166 500/5 = 100 i.e., No. of integer divisibility 5 is 100 is n(C) = 100 Now 500/(2 x 3) = 500/6 = 83.33 divisible 2 and 3 is 83. n(A ∩ B) = 83. We have n(A ∩ B ∩ C) = n(A) + n(B) + n(c) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C) = 250 + 166 + 100 – 83 – 33 – 50 + 17 = 533 – 166 = 367Read more on Sarthaks.com - https://www.sarthaks.com/599638/how-many-integers-between-1-and-500-are-divisible-by-2-3-or-5

Answer 2

Given :   the number of integers between 1 and 500 that are not divisible by any of the integers 2,3 and 5

To Find : Count of numbers

Solution:

the number of integers between 1 and 500  are from

2 to 499   ( total Integers are )  498

Divisible by 2  

2 , 4 , 6 , ... , 498       ( 249  integers)

Divisible by 3

3 , 6 , 9 , ... ,  498     ( 166  integers)

Divisible by 5

5 , 10 , 15, ..   . 495   ( 99 integers)

Divisible by 6

6 , 12 , ... ,  498  ( 83 integers)

Divisible by 10

10 , 20 , ... ,  490  ( 49 integers)

Divisible by 15

15 , 30 , ... ,  495  ( 33 integers)

Divisible by 30

30 , 60 , ... ,     480  ( 16 integers)

number of integers between 1 and 500 that are divisible by any of the integers 2,3 and 5

= 249 + 166 + 99 - 83 - 49 - 33 + 16

= 365

Number that are not divisible by  any of the integers 2,3 and 5

are 498 - 365 = 133

133  integers between 1 and 500 that are not divisible by any of the integers 2,3 and 5

if integers 1 and 500 are also included then 1 will also be added ( 500 will not be added)

then 133 +  1 = 134

134  integers from 1 and 500 that are not divisible by any of the integers 2,3 and 5

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